Convergence of gradient observer for rotor position and magnet flux estimation of permanent magnet synchronous motors

In Bernard and Praly (2017), we introduced a new sensorless rotor position observer for permanent magnet synchronous motors which does not require the knowledge of the magnet's flux : only electrical measurements and knowledge of the resistance and inductance are needed. In fact, this observer extends the gradient observer from Lee et al. (2010) with the estimation of the magnet's flux. In this paper, we prove its asymptotic stability provided the voltages/intensities (and some of their derivatives) are bounded, and the rotation speed remains away from zero. The proof relies on finding appropriate changes of coordinates allowing the construction of a weak Lyapunov function by backstepping, and the study of its invariant sets.